'Pack into Blend' doesn't include .Mov Files
Mov files can't currently be packed

As said in the manual at the Packed Data page:


  Some typically heavy external files, like video ones from the Sequence
  Editor or Movie Clips, cannot be packed in a blend-file.


Your file fall into the Movie clips category, which unfortunately has been exclude from the files format that can be packed. You'll have to copy between folders and update the path.

If you are using an "image as plane", you should browse in the shader editor the image sequence node and change the path from the Open Image icon: